The present invention relates generally to a precisely controlling a harmonic drive and more particularly to a system for compensating for kinematic error in harmonic drives. Still more particularly, the present invention relates to an error algorithm for harmonic drives that takes into account both static and dynamic components of kinematic error.
Harmonic drives are special flexible gear transmission systems. A typical harmonic drive 10 as shown in FIG. 1, comprises a wave generator 12, a flexible race ball bearing (not shown), a flexible spline 14, and a circular spline 16. Wave generator 12 is a rigid steel core having an elliptical shape with a very small but non-zero eccentricity. Wave generator 12 is surrounded by the flexible race ball bearing. The flexible spline 14 (or xe2x80x9cflexsplinexe2x80x9d) is a thin-walled hollow cup preferably of alloy steel. External gear teeth 13 are machined at the open end of this cup and the closed end is connected to an output shaft. Circular spline 16 is a rigid internal gear having two teeth more than the number of teeth on flexspline 14. When assembled, wave generator 12 fits into the open end of the flexspline cup and gives it an elliptical shape at that end. The teeth of circular spline 16 then mesh with the flexspline teeth at the major axis of the ellipse defined by the wave generator. A fully assembled harmonic drive is shown in FIG. 2. In the most common speed reduction configuration, wave generator 12 serves as the input port, flexspline 14 acts as an output port, and circular spline 16 is held immobile.
This typical construction, with meshing at two diametrically opposite ends, gives harmonic drives many useful characteristics. These include compact design with less weight, higher gear reduction with almost zero backlash, and higher torque-to-weight ratio. Hence, these drives are popular in many precision positioning applications, such as in wafer handling machines in the semiconductor industry, in lens grinding machines, and in rotary adjustment mechanisms of reconnaissance cameras. Harmonic drives are also ideal for space robots because of their higher torque-to-weight ratio, which enables them to be directly mounted at robot joints. Additionally, they are widely used in precision measuring devices and in the semiconductor industry for laser mirror positioning.
The concept of harmonic drives was conceived and developed during the mid-1950s. Their industrial use in different applications has been growing since then. However, the research in the theoretical aspects of their transmission characteristics has not been extensive. Most of the work in this area has addressed nonlinear transmission attributes including kinematic error, flexibility, and hysteresis, and in design attributes including tooth stresses and geometry.
Of the different transmission attributes mentioned above, kinematic error is of foremost concern for precision positioning applications. The kinematic error xcex8K is defined as the deviation between the expected output position and the actual output position. It is illustrated in FIG. 3 and given by the following equation
{tilde over (xcex8)}=xcex8m/Nxe2x88x92xcex8J3xe2x80x83xe2x80x83(1)
where xcex8m is the rotational position of the motor shaft attached to the wave generator, N is the gear reduction ratio, and xcex8l is the rotational position of the output shaft connected to the flexspline or the circular spline as the case may be. The experimental kinematic error waveforms presented in the literature show small magnitude with periodic nature; for instance, the waveform shown in FIG. 4 is periodic with magnitude of 0.05 deg. Also, the fundamental frequency of these waveforms is reported to be twice the frequency of wave generator rotation. In addition to the fundamental error, small high frequency error components are observed. Besides producing a static error in load position, kinematic error acts as a periodic exciter and causes undesirable vibration effects. These vibrations serve as an energy sink and produce dramatic torque losses and velocity fluctuations. Thus, kinematic error has both static and dynamic effects, which lead to performance degradation in both precision regulation and tracking. Hence, compensation of kinematic error is of utmost importance for precision positioning with harmonic drives.
Properties of kinematic error and causes of its occurrence have been studied in the past by several researchers, but a complete characterization of kinematic error has been done only recently. As set out in xe2x80x9cOn the Kinematic Error in Harmonic Drive Gearsxe2x80x9d, Ghorbel et al., which is appended hereto and incorporated herein in its entirety, it has been found that kinematic error differs for different drives, speeds, assemblies, and loading conditions. In particular, as motor speed is increased, the kinematic error waveform is colored by flexibility effects. This leads to the concept of a xe2x80x9cpure formxe2x80x9d of kinematic error. The pure form, defined at a low speed, varies as the load on output shaft is increased. Also, this form changes with the change in assembly conditions. In addition, kinematic error has been reported to be sensitive to the environmental conditions. Thus, compensation for this nonlinear, operating condition-dependent, drive-specific kinematic error poses a challenging task.
Heretofore, the complete compensation for kinematic error in set-point and trajectory tracking with harmonic drives not heretofore been achieved. One previous attempt to compensate for kinematic error constituted approximating the kinematic error with a simple sinusoid
{overscore (xcex8)}=A sin 2xcex8m.xe2x80x83xe2x80x83(2)
by neglecting the higher frequency components in the error and determining the trajectory to be traversed by the motor xcex8m(t) was determined using equation (1) for a given load trajectory xcex8l(t). Next, the motor position was controlled to follow the trajectory xcex8m(t), thereby partially compensating for the kinematic error when the output tracked the trajectory xcex8l(t). This scheme required prior knowledge of the error waveform, and it did not account for the error sensitivity to different factors mentioned above. This compensation can be considered open loop because no load side feedback is used.
The second previously known approach is an active compensation approach, which differs from the open loop approach described above in that the disturbance is injected as the current controller input to compensate for the error. Before implementing this scheme, the disturbance injection signal has to be calibrated based on the kinematic error profile, for which the measurement of acceleration at the output is used. Thus, this scheme, too, requires prior knowledge of the kinematic error in a different way. Additionally, once calibrated, this scheme does not ensure complete compensation if the assembly, loading or environmental conditions change. Thus, both these approaches seek to compensate for the error in an open loop sense, using the stored information of the kinematic error.
The present invention includes the use of control algorithms in controllers to completely compensate for kinematic error without using prior information of the error. The present algorithms are based on a mathematical model that represents nonlinear dynamic effects of kinematic error in harmonic drives. A general form of kinematic error is assumed while deriving the equations of dynamics using the Lagrange formulation. With reference to this model, the present nonlinear control algorithms compensate completely for the kinematic error both in set-point and trajectory tracking.
The asymptotic stability of error dynamics equilibrium with the present controllers is demonstrated using the Lyapunov theory. Simulation and the experimental results obtained using a dedicated harmonic drive test setup verify the effectiveness of these controllers. The present controllers achieve the compensation task independent of the form of kinematic error and no prior information regarding the form is necessary. Instead, the present control scheme uses both load side and motor side feedback. Hence, the present compensation approach provides complete compensation irrespective of the error form, using a closed loop approach. This is in contrast to previous open loop approaches, which use only motor side feedback.